Collinear points and parallel lines with vectors Problem set B

In triangle PQR, \(\overrightarrow{PQ}=3\mathbf{a}\) and \(\overrightarrow{PR}=2\mathbf{b}\). The point S lies on QR with QS:SR = 1:2. The point T is such that \(\overrightarrow{QT}=\frac{1}{2}\overrightarrow{PR}\). Show that P, S and T lie on a straight line.

In triangle OAB, \(\overrightarrow{OA}=8\mathbf{a}\) and \(\overrightarrow{OB}=5\mathbf{b}\). X is the midpoint of OA and Y is the midpoint of AB. Show that XY is parallel to OB.

Show that the points P(−2, 3, 1), Q(1, 5, −3) and R(4, 7, −7) are collinear.

A roof truss ABC is modelled as a triangle, \(\overrightarrow{AB}=4\mathbf{a}\) and \(\overrightarrow{AC}=3\mathbf{b}\). The point D lies on BC with BD:DC = 3:1. The point E is such that \(\overrightarrow{BE}=3\overrightarrow{AC}\). Show that A, D and E lie on a straight line.

A tabletop KLMN is a trapezium (lengths in decimetres), \(\overrightarrow{KL}=6\mathbf{a}\), \(\overrightarrow{NM}=3\mathbf{a}\) and \(\overrightarrow{KN}=\mathbf{a}+5\mathbf{b}\). H is the midpoint of KL. Show that NH is parallel to ML.

A submarine is detected at three points D(1, −4, −6), E(3, −1, −2) and F(7, 5, 6) (coordinates in metres). Show that the submarine moved in a straight line.

In triangle ABC, \(\overrightarrow{AB}=3\mathbf{a}\) and \(\overrightarrow{AC}=4\mathbf{b}\). The point D lies on BC with BD:DC = 1:2. The point E is such that \(\overrightarrow{BE}=k\overrightarrow{AC}\). Find the value of \(k\) for which A, D and E lie on a straight line.

WXYZ is a quadrilateral, \(\overrightarrow{WX}=10\mathbf{a}\), \(\overrightarrow{ZY}=5\mathbf{a}\) and \(\overrightarrow{YX}=4\mathbf{b}\). The point P is such that WYP is a straight line and Y is the midpoint of WP. Show that WZ is parallel to XP.